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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Calibration of the imaging system (astigmatism method)</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">3D SMLM imaging can be performed by introducing a weak cylindrical
lens into the imaging path to create slight astigmatism in the image
<cite class="ltx_cite">[<a href="#bib.bib14" title="Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy" class="ltx_ref">1</a>]</cite>. This results in images of molecules with different
ellipticity depending on their axial position. When a molecule is
in focus, its image appears round. If the molecule is slightly above
or below the focal plane, its image appears ellipsoidal.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">Calibration of the imaging system is a procedure which determines
the orientation angle <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="15" height="19" alt="\phi"> of the imaged ellipsoids (the camera
chip might not be aligned with cylindrical lens), and the relationship
between the axial position <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi10.png" width="13" height="12" alt="z"> of the molecules and their imaged
widths <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi7.png" width="102" height="21" alt="\sigma_{1}\left(z\right),\sigma_{2}\left(z\right)">. The calibration
is typically performed using a Z-stack of images of sub-diffraction
fluorescent beads. We use a sparse sample with about 10 to 50 beads
in the image and a Z-stack image sequence with an axial range of about
<img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi3.png" width="41" height="19" alt="2\,\mathrm{\mu m}"> and a step size of <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi2.png" width="49" height="16" alt="10\,\mathrm{nm}">.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Determining the orientation angle</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ol id="I1" class="ltx_enumerate">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">1.</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p">A sequence of images from a Z-stack is processed slice-by-slice using
the methods for raw data analysis (<a href="../../filters/ui/Filters.html" title="" class="ltx_ref">image filtering</a>,
<a href="../../detectors/Detectors.html" title="" class="ltx_ref">approximate localization</a>,
<a href="Fitting.html" title="" class="ltx_ref">PSF fitting</a>). Images of the beads are fit independently
using the <a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">elliptical Gaussian PSF</a>
with <span class="ltx_text ltx_markedasmath"><img id="I1.i1.p1.m1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="15" height="19" alt="\phi"></span> as a free parameter.</p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">2.</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p">Results close to circular are discarded as the angle <img id="I1.i2.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="15" height="19" alt="\phi"> cannot
be determined.</p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">3.</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p">The final orientation angle is calculated as the circular mean of
all remaining measurements
</p>
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-25px" src="mi/mi1.png" width="327" height="57" alt="\phi=\frac{1}{4}\atantwo\left(\frac{1}{n}\sum_{i=1}^{n}{\sin\varphi_{i}},\,%
\frac{1}{n}\sum_{i=1}^{n}{\cos\varphi_{i}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">where <img id="I1.i3.p1.m1" class="ltx_Math" style="vertical-align:-8px" src="mi/mi8.png" width="145" height="25" alt="\varphi_{i}=4\left(\phi_{i}\bmod\frac{\pi}{2}\right)"> adjusts
the fitted angles <img id="I1.i3.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi6.png" width="21" height="19" alt="\phi_{i}">, and <img id="I1.i3.p1.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi9.png" width="15" height="12" alt="n"> is the number of measured
beads.</p>
</div>
</li>
</ol>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Ellipticity as a function of an axial position</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ol id="I2" class="ltx_enumerate">
<li id="I2.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">1.</span> 
<div id="I2.i1.p1" class="ltx_para">
<p class="ltx_p">Using the approximate positions of the beads in multiple Z-planes
and the orientation angle <img id="I2.i1.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="15" height="19" alt="\phi">, both determined in the previous
step, the images of the beads are fit again using the <a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">elliptical Gaussian PSF</a>,
but with a fixed angle <img id="I2.i1.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="15" height="19" alt="\phi">.</p>
</div>
</li>
<li id="I2.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">2.</span> 
<div id="I2.i2.p1" class="ltx_para">
<p class="ltx_p">To estimate the coefficients for the <a href="../../calibration/DefocusModels.html" title="" class="ltx_ref">defocusing model</a>,
we first fit the pair of defocusing curves for each bead separately
using an iterative least-squares algorithm which automatically discards
outliers.</p>
</div>
</li>
<li id="I2.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">3.</span> 
<div id="I2.i3.p1" class="ltx_para">
<p class="ltx_p">From the fitted models, we determine a common focal plane of the beads
as <img id="I2.i3.p1.m1" class="ltx_Math" style="vertical-align:-8px" src="mi/mi4.png" width="45" height="25" alt="\frac{c_{1}+c_{2}}{2}"> and shift the data along the <img id="I2.i3.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi10.png" width="13" height="12" alt="z">-axis
such that all beads are positioned at the same focal plane.</p>
</div>
</li>
<li id="I2.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_enumerate">4.</span> 
<div id="I2.i4.p1" class="ltx_para">
<p class="ltx_p">The final coefficients are obtained by fitting the pair of defocusing
curves to all shifted data points. The ‘‘zero’’ axial position is
given by the intersection of the two polynomials.</p>
</div>
</li>
</ol>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Guidelines for the choice of parameters</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<p class="ltx_p">As the signal to noise ratio is usually higher in the 3D calibration
data with fluorescent beads, users should set the threshold, in the
case of the wavelet filter, to 5 to 8 times the standard deviation
of the 1st wavelet level, e.g., <span class="ltx_text ltx_font_typewriter">6*std(Wave.F1)</span>. The rest
of the settings are the same as in 2D data analysis. Use the Preview
button to see detections of the calibration beads with the current
settings.</p>
</div>
</div>
<div id="Sx1.SSx4" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx4.p1" class="ltx_para">
<ul id="I3" class="ltx_itemize">
<li id="I3.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="EllipticGaussianEstimatorUI.html" title="" class="ltx_ref">Rotated elliptical Gaussian function PSF model (3D using astigmatism)</a></p>
</div>
</li>
<li id="I3.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I3.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="../../calibration/DefocusModels.html" title="" class="ltx_ref">Defocusing models</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib14" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">B. Huang, W. Wang, M. Bates and X. Zhuang</span><span class="ltx_text ltx_bib_year">(2008)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Science</span> <span class="ltx_text ltx_bib_volume">319</span> (<span class="ltx_text ltx_bib_number">5864</span>), <span class="ltx_text ltx_bib_pages"> pp. 810–3</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1126/science.1153529" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Calibration of the imaging system (astigmatism method)" class="ltx_ref"><span class="ltx_text ltx_ref_title">Calibration of the imaging system (astigmatism method)</span></a>.
</span>
</li>
</ul>
</div>
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